Can you please answer the following MCQ showing any equations and steps? 1- If a
ID: 3131849 • Letter: C
Question
Can you please answer the following MCQ showing any equations and steps?
1- If a random variable Y follows the binomial distribution, and the variance of the number of successes in 50 trials is 12, what is the expected value of Y in these 50 trials ?
(a) 7.22
(b) 7.2
(c) 20
(d) More than one expected value of Y is possible based on the information given.
2-The number of mangoes falling from the trees on a tropical island in a single day in summer is a Poisson random variable. The expected number to fall per summer day is 5.2.
What is the probability that 5 mangoes fall in a particular day in summer, correct to 3 decimal places?
(a) 0.500
(b) 0.962
(c) 0.175
(d) Not enough information given to answer the question.
3- In a large sack there are 9 black spheres and 6 blue spheres. If 5 spheres are drawn randomly from the sack without replacement, what is the probability that exactly four of them will be blue, correct to 3 decimal places ?
(a) 0.069
(b) 0.955
(c) 0.045
(d) None of the above.
4- X is the random variable for the number of goals scored in a Premiership football game. Goals are scored at a rate of 2.25 per game. Assuming X follows the Poisson distribution, what is the probability there are three goals scored in each of the next 2 games, correct to 3 decimal places ?
(a) 0.040
(b) 0.019
(c) 0.200
(d) None of the above.
Explanation / Answer
1.
As
Var(X) = np(1-p) = 12
Then if n = 50,
50*p(1-p) = 12
p(1-p) = 0.24
p^ - p - 0.24 = 0
By quadratic formula/solver,
p = 0.6 or 0.4
Hence,
E(x) = n p = 50*0.4 = 20 [ANSWER, C]
[We could have used 0.6 as well, but it wouldn't match any of the choices.]
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2.
Note that the probability of x successes is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 5.2
x = the number of successes = 5
Thus, the probability is
P ( 5 ) = 0.174785003 [ANSWER, C]
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